We will measure for the first time the thermal inertia of the surfaces of the Uranian satellites Oberon and Titania by observing them during and around solar eclipses due to Uranus. The eclipse events of the Uranian satellites in 2007-2008 represent a unique and rare opportunity, occurring every 42 years, to measure their thermal inertia. We will obtain time-resolved Q-band imaging of the satellites using T-ReCS on Gemini South, and model their thermal response to the eclipse event using a rough-surface thermophysical model. The thermal inertia depends on the density, conductivity and heat capacity of the surface material, and so provides constraints on the composition and physical nature of the surface. In addition, our targets are comparable in size to the icy dwarf-planets Eris, 2003 EL_61, 2005 FY_9, and (perhaps) Sedna. It is not currently possible to place good constraints on the thermal inertia of those objects; our observations afford an opportunity to make that measurement for analogs of those more distant, primitive bodies.